1. Field of the Invention
The present invention relates generally to methods for minimizing the costs of extracting petroleum from underground reservoirs. More specifically, the present invention relates to determining optimal well placement from a three-dimensional model of an underground reservoir.
2. Description of the Related Art
A critical function of reservoir management teams is the generation of a reservoir development plan with a selection of a set of well drilling sites and completion locations that maximizes productivity. Generation of the plan generally begins with a set of reservoir property maps and a set of infrastructure constraints. The team typically includes geologists, geophysicists, and engineers who choose well locations using reservoir models. The wells are located to optimize some desired property of the reservoir that is related to hydrocarbon productivity. In the early development of a field, these models might consist of porosity or lithology maps based primarily on seismic interpretations tied to a few appraisal wells. Once given the model, the team is often asked to quickly propose a set of locations that maximize production. Complicating this endeavor is the requirement that the selected sites obey a set of constraints, e.g. minimum interwell spacing, maximum well length, minimum distance from :fluid contacts or reservoir boundaries, and well configuration constraints. The combined problem is highly combinatorial, and therefore time consuming to solve. This is especially true for reservoirs that are heterogeneous with disconnected pay zones. Practical solutions to this problem typically involve evaluating a small subset of the possible well site combinations as case studies, and then selecting those with the highest value of the desired productivity metric, e.g. net pay or permeability-thickness (represented as xe2x80x9cqualityxe2x80x9d).
As a reservoir is developed with production wells, a more comprehensive reservoir model is built with detailed maps of stratigraphy and pay zones. Pressure distribution maps or maps of fluid saturation from history matching may also become available. Then, proposing step-out or infill wells requires the additional consideration of constraints imposed by performance of the existing wells. Thus, the choice of selecting well locations throughout the development of a reservoir can become increasingly complicated. Again, this is especially true for reservoirs that are heterogeneous with disconnected pay zones. Finding solutions to the progressively-more complex well placement problem can be a tedious, iterative task.
There have been several reported studies that have attempted to use ad hoc rules and mathematical models to determine new well locations and/or well configurations in producing fields. The following publications are hereby incorporated herein by reference:
1. Seifert, D., Lewis, J. J. M., Hern, C. Y., and Steel, N. C. T., xe2x80x9cWell Placement Optimisation and Risking using 3-D Stochastic Reservoir Modelling Techniquesxe2x80x9d, SPE 35520, presented at the NPF/SPE Europeanf Reservoir Modelling Conference, Stavanger, April 1996.
2. P. A. Gutteridge and D. E. Gawith, xe2x80x9cConnected Volume Calibration for Well Path Rankingxe2x80x9d, SPE 35503, European 3D Reservoir Modelling Conference, Stavanger, Apr. 16-17, 1996.
3. Rosenwald, G. W., and Green, D. W., xe2x80x9cA Method for Determining the Optimum Location of Wells in a Reservoir Using Mixed-Integer Programmingxe2x80x9d, SPE J., (1973).
4. Lars Kjellesvik and Geir Johansen, xe2x80x9cUncertainty Analysis of Well Production Potential, Based on Streamline Simulation of Multiple Reservoir Realisationsxe2x80x9d, EAGE/SPE Petroleum Geostatistics Symposium, Toulouse, April 1999.
5. Beckner, B. L. and Song X., xe2x80x9cField Development Planning Using Simulated Annealingxe2x80x94Optimal Economic Well Scheduling and Placementxe2x80x9d, SPE 30650, Annual SPE Technical Conference and Exhibition, Dallas, Oct. 22-25, 1995.
6. Vasantharajan S. and Cullick, A. S., xe2x80x9cWell Site Selection Using Integer Programming Optimizationxe2x80x9d, IAMG Annual Meeting, Barcelona, September 1997.
7. Ierapetritou, M. G., Floudas, C. A., Vasantharajan, S., and Cullick, A. S., xe2x80x9cA Decomposition Based Approach for Optimal Location of Vertical Wellsxe2x80x9d, AICHE Journal 45, April, 1999, p. 844-859.
8. K. B. Hird and O. Dubrule, xe2x80x9cQuantification of reservoir Connectivity for Reservoir Description Applicationsxe2x80x9d, SPE 30571, 1995 SPE Annual Technical Conference and Exhibition, Formation Evaluation and Reservoir Geology, Dallas, Tex.
9. C. V. Deutsch, xe2x80x9cFortran Programs for Calculating Connectivity of three-dimensional numerical models and for ranking multiple realizations,xe2x80x9d Computers and Geosciences, 24(1), p. 69-76.
10. Shuck, D. L., and Chien, C. C., xe2x80x9cMethod for optimal placement and orientation of wells for solution miningxe2x80x9d, U.S. Pat. No. 4,249,776, Feb. 10, 1981.
11. Lo, T. S., and Chu, J., xe2x80x9cHydorcarbon reservoior connectivity tool using cells and pay indicatorsxe2x80x9d, U.S. Pat. No. 5,757,663, Mar. 26, 1998.
Seifert et al1 presented a method using geostatistical reservoir models. They performed an exhaustive xe2x80x9cpin cushioningxe2x80x9d search for a large number of candidate trajectories from specified platform locations with a preset radius, inclination angle, well length, and azimuth. Each well trajectory was analyzed statistically with respect to intersected net pay or lithology. The location of candidate wells was not a variable; thus, the procedure finds a statistically local maximum and is not designed to meet multiple-well constraints.
Gutteridge and Gawith2 used a connected volume concept to rank locations in 2D but did not describe the algorithm. They then manually iterated the location and design of wells in the 3D reservoir model. This is a xe2x80x9cgreedyxe2x80x9d approach that does not accommodate the constraints on well locations, and the selection of well sites is done in 2D. Both this and the previous publication are ad hoc approaches to the problem.
Rosenwald and Green3 presented an Integer Programming (IP) formulation to determine the optimum location of a small number of wells. He assumed that a specified production versus time relationship is known for the reservoir and that the potential locations for the new wells are predetermined. The algorithm then selected a specified number of wells from the candidate locations, and determined the proper sequence of rates from the wells.
Kjellesvik and Johansen4 ranked wells"" drainable volumes by use of streamlines for pre-selected sites. The streamlines provide a flow-based indicator of the drainage capability, and although streamline simulation is significantly faster than a full finite-difference simulation, the number of required operations in an optimization scheme, e.g. simulated annealing or genetic algorithm, is still O(N2), where N is the number of active grid cell locations in the model. The compute time is prohibitive when compared with using a static measure. Beckner and Song5 also used flow simulation tied with a global optimization method, but they were only able to perform the optimization on very small data volumes.
Vasanthrajan and Cullick6 presented a solution to the well site selection problem for two-dimensional (2D) reservoir maps as a computationally efficient linear, integer programming (IP) formulation, in which binary variables were used to model the potential well locations. This formulation is unsuitable for three-dimensional :data volumes. A decomposition approach was presented for larger data problems in three-dimensional (3D) maps by Ierapetritou et al7.
Hird and Dubrule8 used flow simulation in 2D reservoir models to assess connectivity between two well locations. This was for relatively small models in 2D and only assesses connectivity between two specific points. C. V. Deutsch9 presents a connectivity algorithm which approaches the problem with nested searches of growing xe2x80x9cshellsxe2x80x9d. This algorithm is infeasibly slow.
Shuck and Chien10 presented an ad hoc well-array placement method that selects the cell pattern of the well-array so that the cell area is customized and the major axis of the cells are parallel to the major axis of transmissivity of the well field. This method does not determine optimal locations for individual wells.
Lo and Chu11 presented a method for estimating total producible volume of a well from a selected well perforation location. No optimization of the total producible volume is sought in this reference.
The above publications fail to provide a feasible method for selecting optimal or near-optimal well completion locations in a 3D reservoir model for a variety of reasons, not the least of which is the size of the problem space. Typical 3D seismic models include 107-108 voxels (volumetric pixels, a.k.a. cells), and the methods described in the above publications cannot efficiently find a solution. Accordingly, a need exists for a systematic method of identifying optimal or near-optimal well locations in a three-dimensional reservoir model. Preferably, the method would be computationally efficient, and would account for the sophisticated drilling technology available today that allows horizontal and/or highly deviated completions of variable lengths which can connect multiple high-pay locations.
There is disclosed herein a systematic, computationally-efficient, two-stage method for determining well locations in a 3D reservoir model while satisfying various constraints including: minimum interwell spacing, maximum well length, angular limits for deviated completions, and minimum distance from reservoir and fluid boundaries. In the first stage, the wells are placed assuming that the wells can only be vertical. In the second stage, these vertical wells are examined for optimized horizontal and deviated completions. This solution is expedient, yet systematic, and it provides a good first-pass set of well locations and configurations.
The first stage solution formulates the well placement problem as a binary integer programming (BIP) problem which uses a xe2x80x9cset-packingxe2x80x9d approach that exploits the problem structure, strengthens the optimization formulation, and reduces the problem size. Commercial software packages are readily available for solving BIP problems. The second stage sequentially considers the selected vertical completions to determine well trajectories that connect maximum reservoir pay values while honoring configuration constraints including: completion spacing constraints, angular deviation constraints, and maximum length constraints. The parameter to be optimized in both stages is a tortuosity-adjusted reservoir xe2x80x9cqualityxe2x80x9d. The quality is preferably a static measure based on a proxy value such as porosity, net pay, permeabilty, permeability-thickness, or pore volume. These property volumes are generated by standard techniques of seismic data analysis and interpretation, geology and petrophysical interpretation and mapping, and well testing from existing wells. An algorithm is. disclosed for calculating the tortuosity-adjusted quality values.